Advanced Addition and Subtraction

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Grouping, Addition and Subtraction with Beans, Cups, Bowls and Tin Cans

Now that we have completed the basics of beginning addition and subtraction, we continue advancing the skill in this area with the use of a place value board and counters.

Anything can be used as math manipulatives, although these marshmallows proved to be a bit large to fit ten of them in a cup for the tens side. Mini marshmallows worked better..

I begin by having them put a few counters in the ones and a few cups of ten of the same item in the tens column. It really doesn't matter what you start with. Then I have him subtract one at a time from the ones column until he runs out of ones. He now has to dump the contents of the tens cup on the ones side of the board. This can be played in reverse too, adding one at a time. When ten are reached on the ones column, they are scooped up and put in a cup to be placed in the tens column.

We play this plus-one and minus-one game a few times.

We then play the game again, recording on strips of paper how many items (ones) and how many cups (tens) are in each step of the game. We examine the columns of figures for possible patterns. Can you predict the next number in the series? You can also play the minus-one game and record the patterns. Are the patterns the same for the plus-one and the minus-one games?

At some point bowls can be added for a hundreds column. We run up and down the amounts with these as well, noting patterns. Are they the same? They soon find out that one can avoid lengthy counting by noticing the patterns within the numbers. Understanding place value they begin to understand helps them to know the amounts without having to count every bean.

At this point I need to take some time out to make sure he knows enough about place value that he can accurately identify the larger and smaller of two numbers so that when he make up his own additon and subtraction problems, he can be sure to put the larger number and small number in the correct places.

 So now we play a place value game, which also covers less than/more than symbols to indicate who won.

I favor working with beans, cups and bowls on a place value board as a beginning for all basic math. One day we use it with multi-digit addition with carrying. This day I gave him the problem 28 +54 = 

and so he set out 2 cups and 8 beans on the board.

and then he added to them 5 cups and four beans. He quickly saw that there were more beans than available spots in the beans, or ones column and so he set them to the side.

I had him fill up the ones column...

put them into a cup (or trade them for an already filled cup) and then put the remaining two into the beans/ones column.

We then filled in the answer on a chart. He will do this hands-on regrouping for a time and then when he feels he wants to fill out the chart without using the beans, I will show him the more traditional way of carrying. By this time, it will seem obvious or second-nature to carry, or change the ones into tens as needed.

Next we add a tin-can for the thousands column and we roll the die to make up addition problems. For the first row, we roll for times for the four place values ones-thousands. Next we roll three times for the next row. Now we add (or subtract) them.

We begin recording these problems, too.

Grouping, Addition and Subtraction with Chips
The chips to which they will now work with provide the necessary link between concrete representation of beans and cups and the more abstract numbers representing unseen quantities.

We begin with chip trading in order for him to learn that each chip, as we progress to the left, represents ten times the chip to the right.

We begin by his putting seven chips in the green and blue columns and we add one, two or three to the green column, and once he get ten greens he trades them in for a blue chip.

He also trades ten blue chips, when he gets them, for one red chip. We play this game until he can easily trade back and forth when adding and subtracting one to three chips at a time. It is sort of a banking game.

Once he proficient at this, he can begin adding and subtracting, by the roll of four die for the top amount and three for what is added or subtracted.

When he feels comfortable doing so, he can begin to create his own problems in base ten.
He can also learn that he can verify his addition answers by subtracting and verify his subtraction problems by addition.

"Students must decide when, if ever, they wish to abandon aids to computation." -Mathematics; a Way of Thinking, Bob Baratta-Lorton

source: Mathematics; a Way of Thinking, Bob Baratta-Lorton

Begninning Addition and Subtraction

"The ability to think is the most valuable skill we can pass on to our students, for more important than any particulars of knowledge. If our goal is to let students think for themselves, we must constantly change how we pose problems"

"We can provide students with the power to know, and to know when they know, and we must help them make full use of this power. Whatever abilities we are able to instill in them will remain with them -we won't."

I start by getting them to take a handful of tiles and put them on a pile in front of themselves on the table and count them. Then I ask them to take a second pile, keeping this pile separate from the first pile. Count how many are in that pile.
Push the two piles together and count how many there are all together.

Once we have done this a number of times, they begin recording them.
He now is creating his own addition problems. He is free to create problems of any size that he wishes. He can advance at any rate he feels comfortable. I tell him that if he has created enough two-pile problems, he can begin making up problems that use three handfuls. There is no special praise or no pressure to advance to the next level, so he can form his own contentment at his own level of success.
The suggestions of other things to do are just to keep the interest alive. Adding with three, four of five piles are just variations on the same theme. Suggesting that he predict how many there are all together can also be another variation.

"The more we allow our students to rely on their own intelligence in finding answers to problems, the less difficult teaching mathematics or any other subject becomes."


There is no guarantee that he can count accurately enough to produce correct answers to the problems created, but it is more likely that the answers will be correct if the emphasis is on the making and doing of the problems and not the finishing of the page or chapter or book. He begins to own and take pride for his problems and his answers.

After he seems to tire of addition problems, I introduce subtraction problems in a similar way. I get him to take a large handful of tiles and put them in front of him. Count them. This time instead of taking another handful of tiles, I get him to take some of the tiles in front of him into his hand. Count these and put them back in the box of tiles. Count the ones remaining. Once this procedure is learned, he can begin recording.
First list the whole pile, then the ones taken away and held in the hand, a line and then those remaining.
Once he tires of making and recording subtraction problems, then I ask him to make up addition and subtraction problems alternately. If it has not come up before, we talk about the difficulty in telling right away whether they are addition or subtraction problems and use this opportunity to teach the use of plus and minus symbols.

Once he has done this for a period of time, I suggest a game in which we take a number, 9 for example, and add 2 more, and then keep adding 2 more and recording the numbers. We then look at the answers for patterns. Do the patterns repeat themselves? Are they odd or even numbers? How many numbers are in the right-hand column before the pattern repeats itself?
These can be done for a while, choosing different sets of numbers of his choice.

Quentin wanted to add some multiplication and division problems with 5 as well, so we did this, too.

Once we tire of this, we do another game in which I pick a number and he tells me what the problem could have been to get that particular answer. Can you come up with a different problem that could get this answer? Is 5 plus 3 the same as 3 plus 5? These can also be written down.

Once we tire of these games with the materials we have been using, I bring out different material. He then gets to see that all that he has learned about tiles can transfer to cubes and that it is not tied to the materials themselves. All of this is done at his own pace.

(These activities, of course, are not all done in one sitting, but over several months of sittings. This  method of teaching and all quotes are from Mathematics...A Way of Thinking, Robert Baratta-Lorton)

For more advanced work, go to Advanced Addition and Subtraction.

Place Value with Marshmallows

If you have been looking at our learning activities for any time, you already know that we love to include treats and food as a part of our schooling. It draws them in and makes whatever activity we are doing a little more like a celebration, and that is the way learning should feel, I think. Today I reviewed place value with Quentin using marshmallows. We had three sizes of them..the mini's, the regular-sized and those jumbo-sized ones that are all the rage right now.  Perfect for showing 1's, 10's and 100's places.

This idea comes from Relentlessly Fun, Deceptively Educational, and if you have not gone to this blog before, you must go there and check it out. I find myself bookmarking every post she does!

We started out with him just building various numbers, and then after awhile we began doing some three digit addition problem without regrouping.

 Then we got the idea of putting the mini's on a wooden skewer -sticky business, but he seemed to enjoy it anyway. Then we started with addition problems with some regrouping. Once the skewer held 10 mini's, he traded it in for a regular-sized marshmallow.

It was fun, it held his attention and he was happy to learn.

139 shown in marshmallows

We just used a blank piece of paper for our problems, but if you are the worksheet type, Relentlessly Fun, Deceptively Educational has a great worksheet to go with this lesson which has a nice, colorful key at the top.

Math Games Fit For A King

The Counting House Game at Fairy Dust Teaching

When I saw these lovely math games at Fairy Dust Teaching, I knew that my youngest son would love them. They excite the imagination as well as make an interesting way of practicing math skills. 

To make the boards, we used one 12 x 12 sheet of scrapbook paper per board, a number grid and anything you would like to decorate your board. I used a piece of foam to make the crown. To play the games you will also need a die (we used a 10-sided die for the 100 grid, but Sally at Fairy Dust Teaching, used a 6-sided die for her 20 and 30 grids) and glass counting gems or pennies. Sally also uses peg people.

The Counting House
To play: The child roll a die and counts out this amount of gems and places them on the counting grid. He must count the numbers on the grid as each additional amount of gems are added, until he gets to his final number on the grid. But first, before he plays, tell him a story. Sally has created a lovely story about Mr. Thirty to go with her 30-board. I wanted to use a 100's board, so I created a different story.

Once upon a time there was a King who needed to collect the yearly taxes from his subjects as harvest season had just now ended . So he sent out his men to search the kingdom high and low to find all of the people in the kingdom. They found cottages amid the wood, and houses along the streams. They found dwellings in a little town and those all alone. At each place they collected the taxes and took them to the king's counting house. The people were expecting this because this happened every year this time and the king expected to get 100 coins for that is what he brought in each year. Please check for the King - roll the King's royal dice to see how many coins are collected at each house and fill the Hundred Counting House.

Another variation for older students is for them to keep track of their die rolls and make a very large addition equation.

War of The Kingdoms
You need as many game boards as players, with a minimum of two boards/players. Each game board is a kingdom. The player is the King or Queen of that Kingdom.

To Play: This is a game is just like the The Counting House except  that each player rolls the dice on his turn and then puts that number of markers down on his board. The first player to fill their number board, is the Kingdom that wins!

Joining Kingdoms
This game takes a different board. This board can be a castle or a crown shape, with two circles or squares on it, one large, and one small and a place in which numbers can be put above the boxes. It also requires gems, a 10-sided die (or six, if you wish) and number cards or tiles.

To play:
Roll a die twice to get two different numbers that you will count out markers into the circles on the game board using the "jewels." The number that is less goes into King Less' smaller circle and the number that is more goes into the King More's bigger circle. How many are there when they share and join their jewels together? Next, the child must find that number in the number cards and put it on the joined square. But first before you play, here is my version of the background story (Sally has another version.)

Once upon a time there lived a king who was small in stature (but large in wisdom)  King named King Less and a king who was very tall and broad (but not so bright) named King More. King More kept taking over more and more land and yet he had trouble keeping track of it because he wasn't very bright. King Less kept losing provinces to King More and he kingdom had shrunk to a very small one indeed. His kingdom was a happy one, however, because he was kind and managed his kingdom well.  One day, as King More was taking over more villages to his kingdom, he saw that the people in King Less' kingdom were much happier than in his own so he decided to go to the King Less to find out how he could be have more happy kingdom. King Less told him, "I will teach you the secret to having a happy kingdom if you can help me defend my kingdom from invaders on all sides - then we both will have more!"  That day King Less and King More joined their kingdoms and became co-rulers. King Less, although he was small and brought less land to their union, used his wisdom to run the kingdom within, while King More used his strength and the wealth of all his land to defend the boarders of their kingdom. Together they had a very happy and strong kingdom.

Laminate boards for durability.  These games should be played with adult supervision and are intended for children 5 years old or older as they use small pieces on the game boards.  Please adjust for safety if you are making these for younger children.

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Mayans and Toltecs (500-1500) and Base 20 Math

The Maya people lived in small kingdoms in the rainforests and plains of Central America. Their temples were pyramids and they played a fast-moving ball game. They had a very accurate 365 day calendar.

Around 1160 fierce tribes began invading, scattering both the Toltecs and the Mayas.
Around 850m most of the Maya had moved north into the area known as the Yucatan. Meanwhile, in Mexico the Toltecs became powerful. In the 1100's invadig tribes drove the Toltecs out of Mexican and into the Yucatan, where it seems as if they worked together peacefully.

Remember the fun we had with base 6 math? The Mayan's math system was a base 20 system.

So, I got out the counters and we decided to have some fun with Mayan Math.

First we made up a name for the numbers from 10- 20, because it was easier to understand the place value if we used differnt names for the digits that we have in our base 10 that would not also be in base 20. So, our ten became bleep...11 became cheep...12 became seep and so on to 20.

 Then we went through the same games as we had gone through with our base 6 math.
It was confusing and fun and we laughed and got frustrated. I was so busy I forgot to take photos. 

We looked again for patterns.

We tried adding and subtracting using base 20. I was hard but fun.

Place value is beginning to take on new meaning.

 In 1542 the Mayans surrendered to Francisco de Montejo.

More Activities

Another great Mayan activity to do is this Jade Mask at Academia Celestia.
Makita has a lot of great links there, too.
You could also make Mayan worry dolls from the tutorial at Child in Harmony which are still made in Guatemala.